Inner Automorphism - Knowlpedia

Let $G$ be a group. For each $g\in G$ , the map

c_g:G\to G,\quad c_g(x)=gxg^{-1}

is an automorphism, called the inner automorphism determined by $g$ . The set of all inner automorphisms is a subgroup $\operatorname{Inn}(G)\le \operatorname{Aut}(G)$ of the automorphism group.

Remarks

The assignment $g\mapsto c_g$ is a homomorphism $G\to \operatorname{Inn}(G)$ whose kernel is the center. Hence there is a natural isomorphism

\operatorname{Inn}(G)\cong G/Z(G),

a quotient group measuring how far $G$ is from being abelian.

Examples

If $G$ is abelian, then $c_g=\mathrm{id}_G$ for all $g$ , so $\operatorname{Inn}(G)$ is trivial.
In $S_3$ , the center is trivial, so $\operatorname{Inn}(S_3)\cong S_3$ .
Inner automorphisms are exactly the permutations of $G$ arising from the conjugation action.